In this talk, I would introduce spontaneous nematicity in the background of fractional quantum Hall fluids where symmetry breaking phenomenon intertwined with topological phase of matter. The resulting nematic FQH state is characterized by an order parameter that represents these quadrupolar fluctuations, which play the role of fluctuations of the local geometry of the quantum fluid.

Physical properties around quantum critical points (QCPs) are of extensive current interests because the fierce competition between critical quantum and thermal fluctuations near the QCPs can strongly affect dynamics and thermodynamics, leading to unconventional physics.

How does thermalization in quantum systems work? Naively, the unitary time evolution prevents thermalization, but one can easily show that in general quantum systems thermalize when brought into contact with a thermal bath. In noninteracting systems, the approach to the thermal value can be either ballistic or diffusive depending on particle statistics and bath temperature.

However, many systems cannot be thermalized when placed in a bath: glasses.

The frequency-dependent longitudinal and Hall conductivities — σ_xx and σ_xy — are dimensionless functions of ω/T in 2+1 dimensional CFTs at nonzero temperature. These functions characterize the spectrum of charged excitations of the theory and are basic experimental observables. We compute these conductivities for large N Chern-Simons theory with fermion matter. The computation is exact in the ’t Hooft coupling λ at N = ∞.

When the wavefunction of a macroscopic system unitarily evolves from a low-entropy initial state, there is good circumstantial evidence it develops "branches", i.e., a decomposition into orthogonal components that can't be distinguished from the corresponding incoherent mixture by feasible observations, with each component a simultaneous eigenstate of preferred macroscopic observables. Is this decomposition unique? Can the number of branches decrease in time?

We consider d=2 fermions at finite density coupled to a critical boson. In the quenched or Bloch-Nordsieck approximation, where one takes the limit of fermion flavors N_f→0, the fermion spectral function can be determined {exactly}. We show that one can obtain this non-perturbative answer thanks to a specific identity of fermionic two-point functions in the planar local patch approximation. The resulting spectrum is that of a non-Fermi liquid: quasiparticles are not part of the exact fermionic excitation spectrum of the theory.

Dimer models have long been a fruitful playground for understanding topological physics. We introduce a new class -- termed Majorana-dimer models -- where the dimers represent pairs of Majorana modes. We find that the simplest examples of such systems realize an intriguing, intrinsically fermionic phase of matter that can be viewed as the product of a chiral Ising theory, which hosts deconfined non-Abelian Ising quasiparticles, and a topological (p − ip) superconductor.

Quantum many-body systems are challenging to study because of their exponentially large Hilbert spaces, but at the same time they are an area for exciting new physics due to the effects of interactions between particles. For theoretical purposes, it is convenient to know if such systems can be expressed in a simpler way in terms of some nearly-free quasiparticles, or more generally if one can construct a large set of operators that approximately commute with the system’s Hamiltonian.